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Home , David Rittenhouse, Extreme value theorem, Johann Heinrich Lambert, John Napier

The German Mathematical Impact: Leibniz, , and Beyond

Germany has contributed to the development of for hundreds of years. Because of the German influence on the city of Milwaukee and the state of Wisconsin, we thought that it would be interesting to examine some of these achievements. Although Germany did not become a unified nation until 1871, the origin of the country began with Charlemagne in the ninth century. Following his death in 814, his kingdom was divided into three parts, one of which corresponded approximately to the modern German state. Mathematics began to blossom there during the Renaissance, in part because the of the printing press made scholarly works much more accessible.

Let us start with the best of the fifteenth century, Johann Müller (1436-1476) of Königsberg. Müller adopted the name Regiomontanus, the Latinized version of Königsberg, which translates into English as King’s Mountain. as a separate branch of mathematics was inaugurated by Hipparchus (ca. 190 BC-ca. 120 BC) in ancient Greece to support his work in and extended by Claudius Ptolemy (ca. 100-ca. 170) in his book Almagest. Although Regiomontanus had access to Latin translations of the Almagest, he realized that Ptolemy’s discussion of trigonometric results was largely ad hoc and that a systematic development of the subject was needed. He established it as a mathematical discipline independent of astronomy by providing an extensive account of methods for solving triangles and by proving the law of sines. His book, On Triangles of Every Kind, discussed both plane and spherical trigonometry.

Regiomontanus had broad mathematical and scientific interests. In addition to his contributions to trigonometry, he worked on geometric construction problems and reform of the Julian calendar. He established an observatory and a printing press at Nuremberg, where he wrote on astronomy and planned to issue translations of the works of Archimedes, Apollonius, Heron, Ptolemy, and others. However, his premature death terminated the project. s Works on arithmetic and algebra also became available in Germany, including one by Johann Widman (b. ca. 1460), who first used the modern symbols for plus and minus. Another by Christoph Rudolff (ca. 1500-ca.1545) employed fractions and the modern notation for roots, and a third by Michael Stifel (ca. 1487- 1567) made extensive use of negative coefficients although he rejected negative solutions of equations. His work, Arithmetica Integra, was the most important of the German textbooks on algebra, and it influenced other writers for decades. Adam Riese (1492-1559), popularized computation using the Hindu-Arabic system so effectively that the phrase, nach Adam Riese, i. e., according to Adam Riese, is used to this day in Germany to attest to the accuracy of a calculation.

The influence of Stifel’s text extended beyond Germany by becoming the basis for the first English book on algebra. Published by Robert Recorde (ca. 1512-1558) in 1557, The Whetstone of Witte encouraged the study of mathematics in England

1 during the Renaissance. He also published an abridgement of Euclid’s Elements, which was the first English work on geometry, and one on astronomy, which advocated the Copernican theory of the solar system. Another of his contributions was the modern symbol for equality. He devised it, he said, because he could think of nothing more equal than two parallel lines.

Trigonometry received further stimulus from the German mathematician, Georg Joachim Rheticus (1514-1576), a student of Nicolaus Copernicus (1473-1543), who did extensive trigonometric work to support his investigations in astronomy. Rheticus published the initial account of Copernicus’ heliocentric theory of planetary motion, and he was the first mathematician to define the trigonometric ratios in terms of the sides of a right triangle. He made full use of all six functions because he calculated tables for each of them.

One of the Renaissance’s most original contributions to mathematics was the study of perspective, motivated by Italian painters who wanted to develop techniques of projecting a three-dimensional objects onto a two-dimensional surface. The first artist to consider this question seriously was Filippo Brunelleschi (1377-1446), and another, Leon Battista Alberti (1404-1472), wrote the first book on the subject, in which he commented that the most fundamental requirement of a painter is to know geometry. Leonardo da Vinci (1451-1519) made a similar observation: “Let no one who is not a mathematician read my works.” Yet another artist, Piero della Francesca (1420-1492), advanced Alberti’s investigation still further in a book in which he thoroughly discussed the mathematical basis of painting. The interest in perspective spread to Germany because of Albrecht Dürer (1471-1528), who studied for several years in Italy. Dürer had a strong interest in the relationship between geometry and art, and he wrote four books on the subject. The concept began to receive a formal mathematical treatment with the Frenchman, Girard Desargues (1591-1661), an and architect, who launched the field of projective geometry.

The concept of projection was important in geography as well as art. At the time of the Renaissance, the most important historical figure in both geography and astronomy was Claudius Ptolemy. Eratosthenes (ca. 276 BC-ca. 194 BC), the librarian at the famous library in Alexandria, founded the scientific study of geography by introducing the concepts of longitude and latitude, which Ptolemy incorporated into the numerous maps he drew. However, as the exploration of the earth expanded dramatically, his work ceased to be adequate. Just as Copernicus revolutionized astronomy by abandoning the Ptolemaic approach, Gerard Mercator (1512-1594), who spent much of his life in the Netherlands but was German by birth, discarded it in geography by introducing the projection which bears his name and which is used to this day.

The idea of a was not introduced until early in the seventeenth century, but there was a hint of it in ancient Greek mathematics. Archimedes wrote a work entitled The Sand-Reckoner, which devised a method of representing extremely

2 large numbers. He was motivated by the need to work with the vast distances encountered in astronomy, and his goal was to show that he could write a number larger than the number of grains of sand necessary to fill the . In the process of developing his system, he noted that numbers could be multiplied by adding what he called their orders. Today we refer to orders as exponents. In effect, he said that numbers could be multiplied by adding their .

Michael Stifel, in addition to his other contributions, noted that there is a relationship between the geometric and the arithmetic sequence 0, 1, 2, 3, . . . . For example, if are multiplied to obtain the exponent 5 of the answer is the sum of the corresponding numbers, 2 and 3, in the arithmetic sequence. He also observed that the exponent associated with the quotient of two terms in the geometric sequence is the difference of the matching terms in the arithmetic sequence, thereby extending Archimedes’ insight to include division. However, he did not pursue the matter further.

Advances in astronomy during the Renaissance emphasized the need for improved methods of computation. (1550-1617) of read Stifel’s work and applied it to develop the first system of logarithms. Napier’s method was geometric rather than algebraic, and his approach led to logarithmic properties that were somewhat different from the modern ones. He also had no concept of a logarithmic base. Nevertheless, his work created a system essentially equivalent to one with the base of 1/e.

By independently analyzing the problem of complex calculations in the same manner as Napier, Jobst Bürgi (1552-1632), a Swiss watchmaker, effectively developed the system of natural logarithms with its base of e. The English mathematician, Briggs (1561-1639), modified Napier’s work to define the common logarithms we use today.

Napier coined the term “logarithm” to mean “reckoning number,” which emphasized the computational value of the idea. The French mathematician and mathematical , Pierre-Simon de la Laplace, commented that “by shortening labors” logarithms had “doubled the lives of ” by cutting their work in half.

One of the astronomers who benefited from the use of logarithms was Johannes Kepler (1571-1630), whose fundamental goal was to establish the truth of Copernicus’ heliocentric theory by discovering mathematical relationships that govern planetary motion. His efforts culminated in his famous three laws, including the fact that the planets revolve about the in elliptical , not circular ones, as Copernicus had assumed. Kepler also succeeded in calculating the of an ellipse, a result obtained by Archimedes but not known in Europe at the time.

While the initial work on logarithms focused on the methodological aspects of the concept, in the eighteenth century, (1707-1783) made fundamental

3 contributions to the underlying theory. Although Swiss by birth, he spent twenty- five years at the Academy of . In his book, Introduction to Analysis of the Infinite, published in 1748, he defined a logarithmic in terms of the corresponding exponential one and derived its properties from those of the exponential. All subsequent work on logarithms was influenced by Euler’s insights.

During the Age of , which began in the seventeenth century, Germany and other European countries made additional mathematical contributions of profound importance. The person most responsible for inaugurating this period of history scientifically was Isaac (1642-1727). He formulated the fundamental concepts of and applied them to revolutionize the field of . Gottfried Wilhelm Leibniz (1646- 1716), who was born in Leibzig, Germany, developed his ideas concerning the subject with no knowledge of Newton’s work. Like Newton, he understood the techniques of differentiation and integration, and he grasped the inverse relationship between the two. However, the two men approached some aspects of their work differently. For instance, Newton associated integration primarily with antidifferentiation because he focused on physical applications; Leibniz identified it with a process of summation because he addressed geometric questions. Today we would say that Newton emphasized indefinite integration while Leibniz stressed definite integration.

Leibniz was quite adept at devising excellent mathematical symbolism. He is responsible for the notations we employ to denote the differential and the , and he was the first mathematician to use the dot consistently for multiplication. Other innovations of his include the colon for division and the similarity and congruence symbols.

Among his additional accomplishments, Leibniz developed methods of solving separable and homogeneous first order differential equations that still appear in modern textbooks. He introduced the term “function” and used it essentially as we do, solved systems of equations by the method we call Gaussian elimination, formulated the idea of determinants, and discussed the binary number system.

In 1642, Blaise (1623-1662) invented the first mechanical calculating machine, one that would add and subtract. In 1671, Leibniz improved on Pascal’s achievement by designing a device that would also multiply and divide. Their work launched advances in the automation of computation that continue to this day.

Newton’s work on calculus preceded that of Leibniz by about ten years, but the latter published his first. A bitter dispute arose between the two concerning who should receive priority. Today it is recognized, as we mentioned, that they made their discoveries independently and deserve equal credit. The dispute had a very negative effect on British , however, because it isolated them from their counterparts on the European continent during most of the eighteenth century.

The achievements of Newton and Leibniz unleashed a torrent of mathematical and scientific activity, including the work of the Bernoulli family across multiple generations. The most prominent members were the Swiss-born brothers, Jacob (1654-1705) and

4 Johann (1667-1748), both of whom were influenced by Leibniz. The to which they contributed included calculus, differential equations, and probability. In 1696, G. F. A. de L’Hôpital, a student of Johann Bernoulli, wrote the first calculus textbook, in which he discussed the technique that we call L’Hôpital’s Rule. Although L’Hôpital was the first to publish the method, he gave credit to his teacher for discovering it.

Euler was also a student of Johann Bernoulli and became one of the dominant mathematicians of the eighteenth century. Among his vast multitude of achievements was his injection of the concept of a function into the heart of mathematics, including a strictly analytic presentation of the trigonometric functions. Like Leibniz, he also devised fruitful mathematical symbolism, especially functional notation.

Johann Heinrich Lambert (1728-1777), a colleague of Euler at Berlin, recognized similarities between the trigonometric and hyperbolic functions and provided an analytic development of the latter. In the process he introduced the notations, sinh, cosh, and tanh that we use today. Another of his contributions was his proof that π is irrational. He also attempted to derive Euclid’s parallel postulate from the other axioms of Euclidean geometry, but he realized that there was a flaw in his argument. Nevertheless, no other mathematician prior to the nineteenth century came closer to developing a non-Euclidean geometry than he.

The seventeenth and eighteenth centuries were an extraordinarily fertile mathematical period, but many of the advances lacked a logically sound basis. Mathematicians focused primarily on the development of successful techniques, not on the identification of a firm theoretical foundation for their work. Resolving this issue became a major goal of nineteenth century mathematics.

The work of Carl Gauss (1777-1855), who was born in Brunswick, Germany, illustrates this shift in outlook. In 1799, he gave the first proof of the Fundamental Theorem of Algebra. This proposition was initially stated by the French mathematician, Albert Girard (1595-1632), and Descartes (1596-1650) and D’Alembert both tried to prove it. Newton, Euler, and Lagrange also attempted to do so, but none of their efforts was complete. While not fully rigorous by modern standards, Gauss’ demonstration was essentially correct and foreshadowed the emphasis on rigor in the dawning century.

Gauss is ranked with Archimedes and Newton as one of the three greatest mathematicians in history. His Researches in Arithmetic was a landmark publication in number theory, and he established the field of differential geometry, based on initial work by Euler and Gaspard Monge. Other major advances included improving the understanding of complex numbers by defining them as points in the plane and devising the method of least squares.

Unlike Lambert and other predecessors, Gauss realized that mathematicians’ inability to derive the parallel postulate from Euclid’s other axioms suggested the possibility of alternative geometric systems. He became the first mathematician to

5 construct a non-Euclidean geometry by assuming that through a point not on a line, more than one parallel to the line can be drawn. A few years later, John Bolyai (1802-1860) of Hungary and Nikolai Lobachevsky (1793-1856) of Russia independently created the same system. Late in his life, Gauss drew great satisfaction from the fact that one of his students, Bernhard Riemann (1826-1866), developed yet a different geometry in which there are no parallels. In the twentieth century Albert Einstein used Riemannian geometry in his general theory of relativity. Riemann, of course, is also famous for his seminal work on the integral that bears his name.

Developments in calculus provide the most extensive illustration of the emphasis on rigor in the nineteenth century. One of the pioneers in this regard was the mathematician, philosopher, and priest, Bernhard Bolzano (1781-1848), who lived in what is now the Czech Republic. He essentially formulated the concept of a and used it to define continuity and differentiability as we do today. He also clarified the relationship between these latter two ideas. Prior to him, it had been assumed that continuous functions were differentiable throughout their domains. Bolzano was the first to construct an example of a that is nowhere differentiable. He also observed that the convergence of an infinite involves the notion of a limit.

Bolzano’s name is attached to two important theorems. Gauss, in his proof of the Fundamental Theorem of Algebra, used the Intermediate Value Theorem, but he justified it only geometrically, not analytically. Bolzano gave an analytic proof of it by introducing the concept of the least upper bound of a bounded set of real numbers; consequently it is sometimes referred to as Bolzano’s theorem. In addition, he proved what is known today as the Bolzano-Weierstrass theorem, that a bounded infinite set has an accumulation point. Unaware of Bolzano’s work, Karl Weierstrass (1815-1897) later established the same result independently. Because of Bolzano’s contributions to the foundations of calculus, Felix Klein referred to him as “the father of arithmetization.”

The mathematician most responsible for shaping calculus as we know it was the Frenchman, Augustin-Louis Cauchy (1789-1857), whose work had a strong influence on developments in German mathematics. Like Weierstrass, he was unaware of the achievements of Bolzano, who was geographically isolated from the mainstream of European mathematics and whose contributions went unnoticed for several decades. Like Bolzano, he understood that the notion of a limit provided the proper basis for the development of the subject, and he defined the concept rigorously. He expressed limits in terms of the algebraic inequalities that we use today, and his implementation of the idea was so successful that even his epsilon- delta notation has become standardized. He also gave definitions of continuity and differentiability that agreed with Bolzano’s.

Although Cauchy used inequalities when working with the fundamental ideas of calculus, he defined the concepts themselves verbally. Weierstrass employed

6 Cauchy’s epsilon-delta notation to express each definition symbolically. By proving the Extreme Value Theorem, that a continuous function defined on a closed has an absolute maximum and an absolute minimum, he duplicated another of Bolzano’s results unknown to him. In the same vein he also constructed an example of a function that is everywhere continuous but nowhere differentiable. His lectures on calculus at the University of Berlin were so influential that he became known as the “father of modern analysis.”

Sonia Kovalevsky (1850-1891) was born in Russia but moved to Heidelberg in 1868 because as a woman, she could not pursue higher education in her homeland. She took classes from a student of Weierstrass and then went to Berlin to study with the master himself. She was not admitted to the university there, but Weierstrass accepted her as a private student. She worked with him for four years and wrote three important papers, including one on partial differential equations. The University of Göttingen awarded her a degree in 1874, and in 1888, she won a major prize from the French Academy. She became a professor of mathematics at the University of Stockholm and is often considered to be the first professional female mathematician in history.

While teaching calculus, Richard Dedekind (1831-1916), a student of Gauss, realized the need to define real numbers rigorously. The earliest attempt to resolve this issue appears in Book V of Euclid’s Elements and presents the theory of the Greek mathematician, Eudoxus (408-355 BC). The approach is geometrical, not arithmetic, in nature and represented the Greeks’ resolution of the crisis precipitated by the Pythagorean discovery of incommensurable magnitudes. In order to construct his theory, Dedekind arithmetized Eudoxus’ contribution by means of his concept of a Dedekind cut. Other mathematicians were also working on this problem, including Georg Cantor (1845-1918), a student of Weierstrass, who used to offer an alternative explanation. About the same time, he began to develop his theory of infinite sets.

Euclid’s Elements had been admired for centuries as the model of mathematical rigor, but mathematicians were aware of some flaws in its development. Although , the father of the axiomatic method, understood the need for undefined terms, Euclid attempted to define all of his concepts, including “point,” “line,” and “plane.” Also, some of his proofs used ideas not explicitly justified by the postulates and axioms with which he began. The emergence of non-Euclidean geometry focused attention on improving the rigor of the Elements. Several mathematicians of various nationalities, including David Hilbert (1862-1943 and Morris Pasch (1843- 1930) of Germany, Oswald Veblen (1880-1960) and G. D. Birkhoff (1884-1944) of the United States, and Giuseppe Peano (1858-1932) of Italy contributed to this effort. In 1899, Hilbert published an extensive axiomatic development of Euclidean geometry in his work, Foundations of Geometry.

As the twentieth century began, the use of the axiomatic method, coupled with an emphasis on abstraction, gained impetus in Germany and elsewhere. Emmy

7 Noether (1882-1935), who was born in Erlangen, Germany, where her father was a professor of mathematics at the University of Erlangen, studied there and with Hilbert’s support eventually taught at Göttingen. Her particular interest was ring theory, and she gave one of the first axiomatic definitions of that concept. Following Hitler’s rise to power, she was dismissed from her position, but she emigrated to the United States and taught at Bryn Mawr College in . She also lectured at the Institute for Advanced Study in Princeton, New Jersey.

In 1900, the mathematics department at Göttingen was preeminent throughout the world, but the domination of the Nazis severely undermined mathematical activity in Germany and led numerous mathematicians in addition to Noether to flee the country. One of these was Richard Courant (1888-1972), a student of Hilbert, who came to the United States and whose goal was to build the New York University mathematics department, which was of mediocre quality when he joined it, into a second Göttingen. His achievement is known today as the Courant Institute of Mathematical Sciences.

Let us close by noting an interesting mathematical connection between Germany and the United States in the person of David Rittenhouse (1732-1796), who was of German ancestry. Born in Germantown, Pennsylvania, the site of a battle during the American Revolution and today part of , Rittenhouse was a self-taught mathematician and , who read Newton’s Principia as a teenager and taught astronomy at the University of Pennsylvania. As a mathematician, he developed a method of calculating logarithms to the base ten that was equivalent to the use of , of which he was unaware. As an astronomer, he built the first telescope in the United States and used it to chart the movement of the planet . Among his other achievements, he constructed exact scale models of the solar system, two of which are housed today at the University of Pennsylvania and , improved timepieces, and surveyed the boundaries of Pennsylvania and several other states. In 1792, he was appointed the first director of the by .

Rittenhouse was recognized as a leading American scientist during the eighteenth century, second only to , whom he succeeded as president of the American Philosophical Society. The APS was founded by Franklin to promote the acquisition and dissemination of knowledge, and today it is the oldest learned society in the country. From its inception it recruited both domestic and foreign members. Rittenhouse became a member in 1768 and held the offices of librarian, secretary, and vice president, as well as president. Through his work in the organization, he helped to establish an intellectual tradition of open inquiry that in the twentieth century welcomed German and other mathematical and scientific refugees, who, in turn, helped the Allies win World War II. Today there is a square in Philadelphia named in his honor.

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Bibliography

Boyer, Carl. A . New York: John Wiley & Sons, Inc., 1968

Burton, David. The History of Mathematics: An Introduction. 6th Ed. Boston: McGraw Hill, 2007

Eves, Howard. An Introduction to the History of Mathematics. 6th Ed. Philadelphia: Saunders College Publishing, 1990

Katz, Victor. A History of Mathematics: An Introduction. 3rd Ed. Boston: Addison- Wesley, 2009

Kline, Morris. Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press, 1972

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